Maths A - Chapter 12

12

syllabussyllabusrrefefererenceenceStrand:Statistics and probability

Core topic:Exploring and understanding data

In thisIn this chachapterpter12A Informal description of

chance12B Sample space12C Tree diagrams12D Equally likely outcomes12E Using the fundamental

counting principle12F Relative frequency12G Single event probability12H Writing probabilities as

decimals and percentages12I Range of probabilities12J Complementary events

Introduction to probability

MQ Maths A Yr 11 - 12 Page 487 Thursday, July 5, 2001 11:45 AM

488

M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d

Introduction

Sam’s dream is to win lotto and be instantly swept into the millionaire set. Each weekhe submits his six numbers to Q-lotto (an imaginary system) and anxiously awaits theresults of the draw. Many Australians subscribe each week to the various systems, someusing the same numbers, week after week, year after year, hoping one day that theirnumbers will ‘come up’; but what chance does each entry have of winning? How manynumbers, and what combinations of numbers are necessary to win any prize? Will Samwin a million dollars, even if his six numbers do come up? Are there some numbersthat appear to be chosen more frequently than others? Lotto systems are complex. Wewill endeavour to understand a simplified version of them.

In a Q-lotto draw, balls numbered 1 to 45 are placed in a barrel and agitated in orderto mix them thoroughly. Six balls are selected at random, in succession. It is these sixnumbers that entitle an entrant to first prize (the jackpot prize), or a share in first prizeif there is more than one entry with these six correct numbers.

Another two balls are then randomly selected. These two numbers are called

bonus

or

supplementary

numbers. Either of these two supplementary numbers may be usedwith three, four or five of the previous six numbers to give lower level prizes (called

division

prizes). Visit a lotto web site, which will display a table similar to thefollowing one, which advises requirements for the various prize divisions.

Combinations that win

There are five prize divisions in Q-lotto. This table explains what you need to win.

Any potential prize obviously depends on the jackpot size (the total amount ofmoney available to be distributed to all first-prize winners), and the number of winnersin each of the divisions. So what is the chance that Sam will win money in any of thedivisions? A web site will display a table similar to this, showing the chance of winningin any prize division. It appears that Sam’s chance of becoming a millionaire is quiteremote (1 in 8.145 million, in fact)!

Chances of winning

Even if Sam happens to pick the correct six numbers, he will not necessarily win amillion dollars. Examine the following Q-lotto draws for three consecutive weeks. Note

Division Numbers required to win

Division 1Division 2Division 3Division 4Division 5

All 6 winning numbersAny 5 winning numbers plus either supplementaryAny 5 winning numbersAny 4 winning numbersAny 3 winning numbers plus either supplementary

Division Chance based on four games

Division 1Division 2Division 3Division 4Division 5

1 in 8.145 million1 in 679 0001 in 37 0001 in 7331 in 211


C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y

489

that in Week 1, when the jackpot prize was over three million dollars, there were onlythree winners, so each received over one million dollars. In Week 2, the jackpot wastwenty-four million dollars. There were 30 winners, so each winner’s share was less thanone million dollars. The four winners in Week 3 received the same as each of the 30winners in Week 2.

To gain a better understanding of systems such as this, we need to understand thetheory of chance and probability. After considering several simpler examples, we shallreturn to continue our investigation of Sam’s chance of fulfilling his dream.

1

Classify the chance of the following eventsoccurring on a scale of

impossible

to

certain

, then position each on the scale at right.

a

Drawing a black card from a deck of cards

b

Winning the lotto

c

The sun rising tomorrow

d

A 6 will turn up on one roll of a 6-sided die

e

A Head will result if a coin is tossed

f

A total of 15 will result if two 6-sided dice are rolled.

2

Consider rolling a normal 6-sided die.

a

How many faces are on the die?

b

List the numbers on the die.

c

What would be the chance of rolling a 6?

d

How many even numbers are on the die?

e

What would be the chance of rolling an even number?

f

How many of the numbers are prime?

g

What would be the chance of rolling a number which was

not

prime?

Data Draw 1 Draw 2 Draw 3

Date Week 1 Week 2 Week 3

Numbers 20, 30, 38, 40, 41,43

5, 11, 22, 33, 40,45

4, 5, 8, 15, 22, 30

Bonus nos. 1, 29 7, 12 23, 35

No. of winners 3 30 4

Jackpot size $3 600 000 $24 000 000 $3 200 000

6 out of 6 paid: $1 200 000 $800 000 $800 000

5 out of 6 + bonus nos. paid:

$14 000 $10 000 $12 000

5 out of 6 paid: $1300 $1000 $1100

4 out of 6 paid: $40 $350 $30

3 out of 6 + either bonus no. paid:

$25 $20 $20

Impossible Certain


490


3

In a family there are 3 children.

a

Copy and complete the following table to show all possibilities of the order of birthof the children.

b

How many different outcomes are possible?

c

In how many cases are all 3 of the children of the same sex?

d

If there are both girls and boys in the family, how many situations occur where girlsoutnumber boys?

4

Convert the following fractions to decimals (to 3 decimal places, if necessary).

a b c d e

5

Convert the fractions in question

4

to percentages (to 1 decimal place).

6

Calculate the following, giving your answer in both fraction and decimal form.

a

×

×

b

( )

2

c

1

−

d

1

−

( )

2

e

1

−

( )

3

f

( )

2

×

( )

3

Child 1 Child 2 Child 3

Boy Girl Girl

Boy Boy Girl

Terms used in chance

Resources: Pen, paper, newspapers, other print material, other forms of media, World Wide Web or Internet, library.

This activity would be best performed in groups.

We frequently hear words such as:‘There is absolutely no chance . . .’‘I am certain . . .’‘The chance of . . . is very slim.’‘. . . a fifty-fifty chance.’

These terms represent an expression of the chance that a particular event might occur.

Task 1

1

As a group, make a list of as many terms of chance as you can.

2

Rank these terms in order of the likelihood of their occurring (some terms you may consider to be of equivalent chance of occurrence to each other).

3

Draw a line graph and place your terms in an appropriate position on this line.

4

Compare your graph with those compiled by other groups.

56--- 2

3--- 1

6--- 4

52------ 5

12------

12--- 1

2--- 1

2--- 2

3--- 1

8---

14--- 1

2--- 3

4--- 1

4---

inve

stigationinvestigatio

n



491

Informal description of chance

You have booked a ski holiday to Thredbo for the middle of July. What is the chancethat there will be enough snow on the ground for you to ski? There is no exact answerto this question, but by looking at the amount of snow in Thredbo during July over pastyears, we know that there is a very good chance that there will be enough snow to skiagain this year.

We can say that it is very likely that we will be able to ski during July at Thredbo.Terms such as

very likely

,

almost certain

,

unlikely

and

fifty-fifty

are used in everydaylanguage to describe the chance of an

event

occurring. For the purposes of probability,an event is the outcome of an experiment that we are interested in. We can describe anoutcome as a possible result to the probability experiment.

Imagine you are playing a board game and it is your turn to roll the die. To win thegame you need to roll a number less than 7. If you roll one die you must get a numberless than 7. We would describe the chance of this event occurring as certain.

When an event is certain to occur, the probability of that event occurring is 1.

Now let’s consider an impossible situation.In a board game you have one last throw of the die. To win you must roll a 7. We

know that this cannot be done. We would say that this is impossible.

When an event is impossible, the probability of the event is 0.

Task 2The media (printed material, radio, television) often exaggerate events (positively or negatively) in their reports of incidents.

1 Search for and collect statements from a range of media where terms of chance have been used. Comment on the appropriateness of their use.

Task 3The English language has many colourful expressions to describe the chance of an event occurring.

1 Consider the following expressions, and research them through the Word Wide Web and your library, then answer the questions that follow.A ‘That will happen once in a blue moon.’

a What is a blue moon?b How often does a blue moon occur?

B ‘There is Buckley’s chance of that happening.’a Who was Buckley?b How did this saying originate?

2 Investigate to find any similar expressions. What are their origins?


492 M a t h s Q u e s t M a t h s A Ye a r 1 1 f o r Q u e e n s l a n d

The chance of any event occurring will often be somewhere between being certain and impossible and we use a variety of terms to describe where the chance lies in this range as shown in the figure at right.

We use these terms based on our general knowledge of the world, the total possible outcomes and how often an event occurs.

You will need to use terms of chance such as those used above to describe events thatare more likely to occur than others.

The term frequency refers to how often an event occurs. We use our knowledge aboutpossible outcomes to order outcomes from the most frequent to the least frequent.

Very unlikelyImpossible

Unlikely

Fifty-fifty

Probable

CertainAlmost certain

1–2

1

0

SkillSH

EET 12.1

Describe the chance of each of the following events occurring.a Tossing a coin and its landing Heads.b Rolling a 6 with one die.c Winning the lottery.d Selecting a numbered card from a standard deck of playing cards.

THINK WRITE

a There is an equal chance of the coin landing Heads and Tails.

a The chance of tossing a Head is fifty-fifty.

b There is only one chance in six of rolling a 6.

b It is unlikely that you will roll a 6.

c There is only a very small chance of winning the lottery.

c It is very unlikely that you will win the lottery.

d There are more numbered cards than picture cards in a deck.

d It is probable that you will select a numbered card.

1WORKEDExample

Mrs Graham is expecting her baby to be born between July 20 and 26. Is it more likely that her baby will be born on a weekday or a weekend?

THINK WRITE

There are 5 chances that the baby will be born on a weekday and 2 chances that it will be born on a weekend.

It is more likely that Mrs Graham’s baby will be born on a weekday.

2WORKEDExample


C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 493

In the above examples, we have been able to calculate which event is more likely bycounting the number of ways an event may occur. This is not always possible. In somecases we need to use general knowledge to describe the chance of an event occurring.

Consider the following probability problems.‘The letters of the alphabet are written on cards and one card is selected at random.

Which letter has the greatest chance of being chosen, E or Q?’Each letter has an equal chance of being chosen because there is one chance that E

will be chosen and one chance that Q will be chosen.‘Stacey sticks a pin into a page of a book and she writes down the letter nearest to

the pin. Which letter has the greater chance of being chosen, E or Q?’This question is more difficult to answer because each letter does not occur with

equal frequency in written text. However, we know from our experience with theEnglish language that Q will occur much less often than most other letters. We cantherefore say that E has a greater chance than Q of being chosen.

This is an example of using your knowledge of the world to make predictions aboutwhich event is more likely to occur. In this way, we make predictions about everydaythings such as the weather and which football team will win on the weekend.

A card is chosen from a standard deck. List the following outcomes in order from least likely to most likely: selecting a picture card, selecting an Ace, selecting a diamond, selecting a black card.

THINK WRITE

There are 12 picture cards in the deck.There are 4 Aces in the deck.There are 13 diamonds in the deck.There are 26 black cards in the deck. The order of events in ascending order of likeli-

hood:selecting an Aceselecting a picture cardselecting a diamondselecting a black card.

1234

3WORKEDExample

During the 1999 NRL season, the Sydney Roosters won 10 of their first 12 games. In Round 13 they played the Northern Eagles who had won 3 of their first 12 games. Who would be more likely to win?

THINK WRITE

Sydney Roosters have won more games than the Eagles.

Sydney Roosters would be more likely to win, based on their previous results. (Footynote: The Eagles won the game 36–14. Sydney was more likely to win the game but nothing in football is certain.)

4WORKEDExample



This is one example of past results being used to predict future happenings. There aremany other such examples.

Informal description of chance

1 Describe the chance of each of the following events occurring, using an appropriateprobability term.a Selecting a ball with a double-digit number from a bag with balls numbered 1 to

40b Selecting a female student from a class with 23 boys and 7 girlsc Selecting a green marble from a barrel with 40 blue marbles and 30 red marblesd Choosing an odd number from the numbers 1 to 100

2 For each of the events below, describe the chance of it occurring as impossible,unlikely, even chance (fifty-fifty), probable or certain.a Rolling a die and getting a negative numberb Rolling a die and getting a positive numberc Rolling a die and getting an even numberd Selecting a card from a standard deck and getting a red carde Selecting a card from a standard deck and getting a numbered cardf Selecting a card from a standard deck and getting an Aceg Reaching into a moneybox and selecting a 30c pieceh Selecting a blue marble from a bag containing 3 red, 3 green and 6 blue marbles

Weather records show that it has rained on Christmas Day 12 times in the last 80 years. Describe the chance that it will rain on Christmas Day this year.

THINK WRITE

It has rained only 12 times on the last 80 Christmas Days. This is much less than half of all Christmas Days.

It is unlikely that it will rain on Christmas Day this year.

5WORKEDExample

remember1. The chance of an event occurring ranges from being certain to impossible.2. (a) An event that is certain has a probability of 1.

(b) An event that is impossible has a probability of 0.3. There are many terms that we use to describe the chance of an event occurring,

such as improbable, unlikely, fifty-fifty, likely and probable.4. Sometimes we can describe the chance of an event occurring by counting the

possible outcomes, while other times we need to rely on our general knowledge to make such a description.

remember

12AWORKEDExample

1


C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 4953 Give an example of an event which has a probability that could be described as:

4 Is it more likely that a person’s birthday will occur during a school term or during theschool holidays?

5 For each event on the left, state whether it is more likely, less likely or equally likelyto occur than the event on the right.a Fine weather Christmas Day Wet weather Christmas Dayb A coin landing Heads A coin landing Tailsc Rolling a total of 3 with two dice Rolling a total of 7 with two diced Winning a raffle made up of 50 tickets Winning a raffle made up of 200 ticketse Winning a prize in the Lotto draw Not winning a prize in the Lotto draw

6 A die is thrown and the number rolled is noted. List the following events in orderfrom least likely to most likely.Rolling an even numberRolling a number less than 3Rolling a 6Rolling a number greater than 2

7 Write the following events in order from least to most likely.Winning a raffle with 5 tickets out of 30Rolling a die and getting a number less than 3Drawing a green marble from a bag containing 4 red, 5 green and 7 blue marblesSelecting a court card (Ace, King, Queen, Jack) from a standard deckTossing a coin and having it land Heads

8 Before meeting in the cricket World Cup in 1999, Australia had beaten Zimbabwe in10 of the last 11 matches. Who would be more likely to win on this occasion?

9 Which of the following two runners would be expected to win the final of the 100metres at the Olympic Games?Carl Bailey — best time 9.92 s and won his semi-finalBen Christie — best time 10.06 s and 3rd in his semi-finalGive an explanation for your answer.

10

A stack of 26 cards has the letters of the alphabet written on them. Vesna draws a cardfrom that stack. The probability of selecting a card that has a vowel written on it couldbest be described as:

11

Which of the following events is the most likely to occur?A Selecting the first number drawn from a barrel containing 20 numbered marblesB Selecting a diamond from a standard deck of cardsC Winning the lottery with one ticket out of 150 000D Drawing the inside lane in the Olympic 100-metre final with eight runners

a certain b probable c even chanced unlikely e impossible.

A unlikely B even chance C probable D almost certain

WORKEDExample

2

WORKEDExample

3

WORKEDExample

4

mmultiple choiceultiple choice



496


12

The ski season opens on the first weekend of June. At a particular ski resort there hasbeen sufficient snow for skiing on that weekend on 32 of the last 40 years. Which ofthe following statements is true?

A

It is unlikely to snow at the opening of the ski season this year.

B

There is a fifty-fifty chance that it will snow at the opening of the ski season this year.

C

It is probable that it will snow at the opening of the ski season this year.

D

It is certain to snow at the opening of the ski season this year.

13

On a production line, light globes are tested to see how long they will last. Aftertesting 1000 light globes it is found that 960 will burn for more than 1500 hours.Wendy purchases a light globe. Describe the chance of the light globe burning formore than 1500 hours.

14

Of 12 000 new cars sold last year, 1500 had a major mechanical problem during thefirst year. Edwin purchased a new car. Describe the chance of Edwin’s car having amajor mechanical problem in the first year.

15

During an election campaign, 2000 people were asked for their voting preferences.One thousand said that they would vote for the government. If one person is chosen atrandom, describe the chance that he or she would vote for the government.

Sample space

At some time in our lives, most of us have tossed or will toss a coin. Many sports beginwith the toss of a coin.

What is the chance that the coin will land showing Heads? Most people would cor-rectly say fifty-fifty. We need to develop a method of accurately describing the prob-ability of an event.

Before we can calculate probability, we need to be able to list all possible outcomesin a situation. This is called listing the

sample space

. When tossing a coin, the samplespace has two elements: Heads and Tails. To calculate a probability, we need to knowthe number of elements in the sample space and what the elements are.


WORKEDExample

5



In many cases, several elements of the sample space may be the same. In such cases,we can distinguish between the number of elements in the sample space and thenumber of distinct (different) elements.

In some situations there may be more than one element in the sample space that givesus the desired outcome. Favourable outcomes are the elements from the sample spacethat will meet the requirements for an event to occur.

List the sample space for rolling a die.

THINK WRITE

The sample space is the numbers 1 to 6. S = {1, 2, 3, 4, 5, 6}

6WORKEDExample

In a barrel there are 4 red marbles, 5 green marbles and 3 yellow marbles. One marble is drawn from the barrel.a List the sample space.b How many elements are in the sample space?c How many distinct elements are in the sample space?

THINK WRITE

a List each marble in the barrel. a S = {red, red, red, red, green, green, green,S = green, green, yellow, yellow, yellow}

b Count the number of elements in the sample space.

b The sample space has 12 elements.

c Count the number of different elements in the sample space.

c The sample space has 3 distinct elements.

7WORKEDExample

Tegan is playing a board game. To win the game, Tegan must roll a number greater than 2 with one die.a List the sample space. b List the favourable outcomes.

THINK WRITE

a List all possible outcomes for one roll of a die.

a S = {1, 2, 3, 4, 5, 6}

b List all elements of the sample space which are greater than 2.

b E = {3, 4, 5, 6}

8WORKEDExample



Sample space

1 The numbers 1 to 10 are written on cards that are turned face down. The cards areshuffled and one is chosen. List the sample space.

2 For each of the following probability experiments, state the sample space.a Tossing a coinb Rolling a diec The total when rolling two diced Choosing a letter of the alphabete The day of the week on which a baby could be bornf The month in which a person’s birthday falls

3 For each of the following probability experiments, state the number of elements in thesample space.a Choosing a card from a standard deckb Selecting the winner of a 15-horse racec Selecting the first ball drawn in a Lotto draw (The balls are numbered 1–45.)d Drawing a raffle ticket from tickets numbered 1 to 1500e Selecting a number in the range 100 to 1000, inclusivef Drawing a ball from a bag containing 3 yellow, 4 red and 4 blue balls

4 The letters of the word MISSISSIPPI are written on cards and turned face down. Acard is then selected at random.a List the sample space.b How many elements has the sample space?c How many distinct elements in the sample space?

5 A card is to be selected from a standard deck.a How many elements has the sample space?b How many different elements has the sample space if we are interested in:

i the suit of the card?ii the colour of the card?iii the face value of the card?

remember1. The sample space is the list of all possible outcomes in a probability

experiment.2. The number of elements in a sample space is the total number of possible

outcomes.3. In the sample space, there are sometimes several elements that are the same.

We may be asked to count the number of distinct (different) elements in the sample space.

4. Favourable outcomes are the elements from the sample space which meet the requirements for a certain event to occur.

remember

12BWORKEDExample

6

WORKEDExample

7


C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 4996 Jane is playing a game of snakes and ladders. It is her turn to roll the die and to win

she needs a number greater than 4.a List the sample space for this roll of the die.b List the favourable outcomes for this roll of the die.

7 A bag holds 60 black marbles and 40 white marbles. Tony is to choose one of thesemarbles from the bag. Tony wants to select a white marble.a How many elements are in the sample space?b How many favourable outcomes are contained in the sample space?

8

A bag contains 5 blue discs, 9 red discs and 6 yellow discs. To win a game, Jennyneeds to draw a yellow disc from the bag. How many elements are in the samplespace?

9

To win a game Jenny needs to draw a yellow disc from the bag in question 8. Howmany favourable outcomes are there?

10

A raffle has 100 tickets. Chris buys 5 tickets in the raffle. Which of the followingstatements is correct?A There are two elements in the sample space.B There are five favourable outcomes.C There are 100 elements in the sample space.D Both B and C

11 New South Wales are playing Queensland in a State of Origin match.a List the sample space for the possible outcomes of the match.b How many elements are in the sample space?c Is each element of the sample space equally likely to occur? Explain your answer.

12 A bag contains five 20c pieces, four 10c pieces and one 5c piece. A coin is selected atrandom from the bag. Without replacing the first coin, a second coin is then selected.a List the sample space for the first coin selected. How many elements has the

sample space?b Assume that the first coin chosen was a 20c piece. List the sample space for the

second coin chosen.c Assume that the first coin chosen was a 10c piece. List the sample space for the

second coin chosen.d Assume that the first coin chosen was the 5c piece. List the sample space for the

second coin chosen.

13 Write down an example of an event that has 4 elements in the sample space.

14 Write down an example of an event that has 10 elements in the sample space but only4 distinct elements.

A 3 B 6 C 14 D 20

A 3 B 6 C 14 D 20

WORKEDExample

8





500


1

A die is rolled. Describe the chance that the uppermost face is 4.

2

A card is drawn from a standard pack. Describe the chance of selecting a black card.

3

A bag contains four $1 coins and seven $2 coins. Describethe chance that a coin drawn at random from the

bag will be a $2 coin.

4

A barrel containing ballsnumbered 1 to 100 has oneball selected at random fromit. How many elements has thesample space?

5

Five history books, threereference books and ten sportingbooks are arranged on a shelf. Abook is chosen at random from

the shelf. How many elements arein the sample space?

6

For the example in question

5

, howmany distinct elements has the

sample space?

7

For the example in question

5

, ifyou want a sporting book, howmany favourable outcomes arethere?

8

Copy and complete. If an event iscertain, then the probability of it

occurring is .

9

Copy and complete. If an event isimpossible, then the probability of it

occurring is .

10

If Jane needs to select an Ace from astandard deck to win the game, howmany favourable outcomes are there?

1



Matching actual and expected resultsResources: Coin, six-sided die, calculator.

This investigation is best performed in small groups.

Task 1Consider tossing a coin once.

1 List the sample space.

2 Theoretically, if you tossed the coin 60 times, how many Heads and how many Tails would you expect to obtain?

3 In practice, these results may vary. Set up an experiment to determine the outcomes. Within your group, toss a coin 60 times and record the outcomes by copying and completing the table below:

4 How closely do your experimental results match the theoretical ones?

5 Combine your results with those of another group. These figures then represent the tossing of a coin 120 times. How do they compare with what you would expect in theory?

6 Collate and combine the results of all of the groups in your class. Compare these figures with what you would expect in theory. Have you reached any conclusions?

Task 2Repeat the processes in Task 1 using a six-sided die in place of the coin. Tabulate your results, comparing the theoretical and experimental outcomes. What conclusions can you draw?

Task 3Your calculator can be set to generate random integers. If you are unsure how to do this, your teacher will advise you.

1 Tossing a coin can be simulated on a calculator by the random generation of a 1 (representing, say a Tail) or a 2 (representing a Head). Using your calculator, repeat Task 1, simulating tossing a coin 60 times.

2 Set your calculator to generate the random integers 1, 2, 3, 4, 5 or 6. This can be used to simulate the rolling of a die. Repeat Task 2.

inve


n

Outcome Tally

H

T

GCpro

gram

Random

GCpro

gram

Dice 1

GCpro

gram

Coin flip



Computer simulation: Tossing a coin and rolling a die

Resources: Computer spreadsheet.

Task 1This task uses a computer to simulate tossing a coin 60 times. The following instructions refer to the Excel spreadsheet. If you use an alternative spreadsheet your teacher will advise you of the equivalent instructions and formulas.

1 In the spreadsheet, type the entries shown in cells A1, A3, D3, G3, D4, G4, D5 and G5. The figures in H4 and H5 are generated by the results of the experiment. Leave these two cells blank at this stage.

2 Enter the expected results of the experiment in cells E4 and E5.

3 Type in the heading shown in cell A7.

4 The 60 cells in the range A9 to J14 represent the results of the simulation of the 60 coin tosses. In these cells we are going to generate integers 1 or 2 randomly. If a 2 results, we will let that represent a Head and if a 1 results, we will say that represents a Tail. The formula to generate these Heads or Tails randomly is:

=IF(INT(RAND()*2+1)=2,”Head”,”Tail”)Type this formula in cell A9 and copy it to all cells in the range A9 to J14. This will randomly generate a Head or a Tail in each of these cells every time the program is run.

inve


nEXCEL

Spreadsheet

Random number generator

EXCEL

Spreadsheet

Simulating coin tosses DIY

EXCEL

Spreadsheet

Simulating coin tosses



5 The number of Heads and Tails generated are to be counted and the results displayed in cells H4 and H5.In cell H4 enter the formula

=COUNTIF(A9:J14,”Head”)In cell H5 enter the formula

=COUNTIF(A9:J14,”Tail”)

6 The function key F9 causes the computer to simulate the 60 tosses of the coin. Press the key and note the count of the resulting number of Heads and Tails in cells H4 and H5 respectively. Press the F9 key again. Record the results over a number of simulations. How do these figures compare with the theoretical expected results?

Task 2Repeat the previous experiment, simulating the rolling of a die 60 times.

1 Enter the expected numbers of 1s, 2s, etc in cells E4–E9.

2 To generate the integers 1, 2, 3, 4, 5 or 6 randomly in cells A13 to J18,a enter, in cell A13, the formula:

=INT(RAND()*6+1)b Copy the formula to the other cells.

3 To count the numbers of 1s, 2s, etcin cell H4, enter the formula =COUNTIF(A13:J18,1)in cell H5, enter the formula =COUNTIF(A13:J18,2). . . and so on.

4 Run the simulation a number of times (by pressing F9). Record and accumulate the results. Comment on the outcomes.

Task 3Formally record the results of this investigation in report form.

EXCEL

Spreadsheet

Die


504


Tree diagrams

A

multi-stage event

is an event where there is more than one part to the probabilityexperiment. Tree diagrams are used to find the elements in the sample space in a multi-stage probability experiment. Consider the case of tossing two coins. How manyelements are there in the sample space? We draw a tree diagram to develop a systemthat will list the sample space for us.

The

tree diagram

branches out once, for every stage of the probability experiment.At the end of each branch, one element of the sample space is found by following thebranches that lead to that point. All of these elements together give us the outcomes ofthe experiment.

Therefore, when two coins are tossed, the sample space can be written:

S

=

{Head–Head, Head–Tail, Tail–Head, Tail–Tail}

There are four elements in the sample space: Head–Tail and Tail–Head are distinctelements of the sample space.

In many cases, the second branch of the tree diagram will be different from the firstbranch. This occurs in situations such as those outlined in the following workedexamples, where the first event has an influence on the second event. The card chosenfirst can then not be chosen in the second event.

HeadHead

Tail

HeadTail

Tail

1st coin 2nd coinHead–Head

Head–Tail

Tail–Head

Tail–Tail

Outcomes

A coin is tossed and a die is rolled. List all elements of the sample space.

THINK WRITE

Draw the branches for the coin toss.From each branch for the coin toss, draw the branches for the die roll.

List the sample space by following the path to the end of each branch.

S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, S = T5, T6}

12

Head

Tail

Coin toss Die roll123456123456

OutcomesH1H2H3H4H5H6T1T2T3T4T5T6

3

9WORKEDExample



505

Each question must be read carefully, to see if repetition is possible or not. In the aboveexample, the numbers cannot be repeated because we are drawing two cards withoutreplacing the first card. In examples such as tossing two coins, it is possible for thesame outcome on both coins.

When drawing a tree diagram, the tree needs tobranch once for every stage of the experiment. When we roll two dice, there are two levels to the tree diagram. If we were to toss three coins, there would be three levels to the diagram, as shown at right.

The numbers 2, 4, 7 and 8 are written on cards and are chosen to form a two-digit number. List the sample space.

THINK WRITE

Draw the first branch of the tree diagram to show each possible first digit.Draw the second branch of the tree diagram to show each possible second digit. When drawing the second branch, the digit from which the tree branches can’t be repeated.

List the sample space by following the tree to the end of each branch.

S = {24, 27, 28, 42, 47, 48, 72, 74, 78, 82, 84, 87}

Note that in this case, 24 is not the same as 42.

1

2

4

7

8

1st digit 2nd digit478

278

248

247

2

3

4

10WORKEDExample

HeadHead

HeadTailHead

Tail

TailHead

Tail

TailHeadTailHeadTail

1st coin 2nd coin 3rd coin

A coin showing ‘Heads’


506


For a family of four children:a Draw a tree diagram to list all possible combinations of boys and girls.b How many elements are in the sample space?c How many elements of the sample contain 3 boys and a girl?

THINK WRITE

a Draw the tree diagram. a

b List the sample space by following the paths to the end of each branch.

b S = {BBBB, BBBG, BBGB, BBGG, BGBB, BGBG, BGGB, BGGG, GBBB, GBBG, GBGB, GBGG, GGBB, GGBG, GGGB, GGGG}

There are 16 elements in the sample space.

c Count the number of elements that contain 3 boys and 1 girl.

c There are four elements of the sample space which contain 3 boys and 1 girl.

BoyBoy

Boy

Boy

Girl

BoyGirl

BoyGirlBoyGirl

Girl

Girl

Girl

Boy

Girl

Boy

Girl

Boy

Girl

Boy

Girl

BoyGirl

BoyGirl

BoyGirlBoyGirl

2nd child1st child 3rd child 4th child

11WORKEDExample

remember1. A tree diagram is necessary in any example where there is more than one stage

to the probability experiment.2. The tree diagram must branch out once for every stage of the probability

experiment.3. Once the tree is drawn, the sample space is found by following the branches to

each end.

remember



507

Tree diagrams

1

Two coins are tossed. Use a tree diagram to list the sample space.

2

On three red cards, the numbers 1, 2 and 3 are written. On three blue cards, the samenumbers are written. A red card and a blue card are then chosen to form a two-digitnumber. Draw a tree diagram to list the sample space.

3

A family consists of 3 children. Use a tree diagram to list all possible combinations ofboys and girls.

4

A coin is tossed and then a die is rolled.

a

How many elements are in the sample space?

b

Does it make any difference to the sample space if the die is rolled first and thenthe coin is tossed?

5

The digits 1, 3, 4 and 8 are written on cards. Two cards are then chosen to form a two-digit number. List the sample space.

6

Darren, Zeng, Melina, Kate and Susan are on a committee. From among themselves,they must select a chairman and a secretary. The same person cannot hold bothpositions. Use a tree diagram to list the sample space for the different ways the twopositions can be filled.

7

A tennis team consists of six players, three males and three females. The three malesare Andre, Pat and Yevgeny. The three females are Monica, Steffi and Lindsay. A maleand a female must be chosen for a mixed doubles match. Use a tree diagram to list thesample space.

8

Chris, Aminta, Rohin, Levi and Kiri are on a Landcare group. Two of them are torepresent the group on a field trip. Use a tree diagram to list all the different pairs thatcould be chosen. [

Hint

: In this example, a pairing of Chris and Aminta is the same asa pairing of Aminta and Chris.]

9

Four coins are tossed into the air.

a

Draw a tree diagram for this experiment.

b

Use your tree to list the sample space.

c

How many elements have an equal number of Heads and Tails?

10

Three coins are tossed into the air. The number of elements in the sample space is:

11

A two-digit number is formed using the digits 4, 6 and 9. If the same number can berepeated, the number of elements in the sample space is:

12

A two-digit number is formed using the digits 4, 6 and 9. If the same number cannotbe used twice, the number of elements in the sample space is:

A

3

B

6

C

8

D

9

A

3

B

6

C

8

D

9

A

3

B

6

C

8

D

9

12CWORKEDExample

9

WORKEDExample

10

WORKEDExample

11






13 When two coins are tossed there are three elements in the sample space, 2 Heads,2 Tails or 1 Head and 1 Tail. Is this statement correct? Explain why or why not.

14 A two-digit number is to be formed using the digits 2, 5, 7 and 8.a If the same number can be used twice, list the sample space.b If the same number cannot be repeated, list the sample space.

15 The numbers 1, 2, 5 and 8 are written on cards and placed face down.a If two cards are chosen and used to form a two-digit number, how many elements

are in the sample space?b If three cards are chosen and used to form a three-digit number, how many

elements are in the sample space?c How many four-digit numbers can be formed using these digits?

16 A school captain and vice-captain need to be elected. There are five candidates. Thethree female candidates are Tracey, Jenny and Svetlana and the male candidates areRichard and Mushtaq.a Draw a tree diagram to find all possible combinations of captain and vice-captain.b How many elements has the sample space?c If boys are filling both positions, how many elements are there?d If girls are filling both positions, how many elements are there?e If students of the opposite sex fill the positions, how many elements are there?

17 Two dice are rolled.a Use a tree diagram to calculate the number of elements in the sample space.b Steve is interested in the number of elements for each total. Copy and complete the

table below.

c How many elements of the sample space have a two-digit number?

Equally likely outcomesBelow is the field for the 1999 Melbourne Cup.

Source: Courier-Mail 2 November 1999

Total 2 3 4 5 6 7 8 9 10 11 12

No. of elements

MELBOURNE CUP ODDS

HorseLatest odds Horse

Latest odds

1 Tie The Knot2 Central Park3 Sky Heights4 Maridpour5 Arena6 Yavana’s Pace7 The Hind8 Travelmate9 Skybeau

10 The Message11 Streak12 Lahar

10-140-113-480-115-133-110-115-266-1

125-116-166-1

13 Second Coming14 Able Master15 Bohemiath16 Figurehead17 Rogan Josh18 Laebeel19 Brew20 Lady Elsie21 Rebbor22 The Warrior23 Zabuan24 Zazabelle

66-130-125-1

100-15-17-1

33-160-150-150-1

125-180-1


C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 509There are 24 horses in the race. The sample space therefore has 24 elements. How-ever, in this case, each outcome is not equally likely. This is because each horse inthe race is not of equal ability. Some horses have a greater chance of winning thanothers. It is true in many practical situations that each outcome is not equally likelyto occur.

The weather on any day could be wet or fine. Each outcome is not equally likely asthere are many factors to consider, such as the time of year and the current weatherpatterns.

In each probability example, it is important to consider whether or not each outcomeis equally likely to occur. In general, when the selection is made randomly then equallylikely outcomes will result.

In some cases we need to use tree diagrams to calculate if each outcome is equallylikely. A statement may seem logical, but unless further analysis is conducted, wecannot be sure.

In a football match between Brisbane and Parramatta there are three possible outcomes: Brisbane win, Parramatta win and a draw. Is each outcome equally likely? Explain your answer.

THINK WRITE

Each team may not be of equal ability and draws occur less often than one of the teams winning.

Each outcome is not equally likely as the teams may not be of equal ability and draws are fairly uncommon in football.

12WORKEDExample



Two-stage experiments (tossing two coins)

Resources: Pen, paper, two coins, calculator, computer spreadsheet.

This investigation would be best performed in pairs.

Task 1Consider the theoretical tossing of two coins.1 Draw a tree diagram to show all possible outcomes.2 List the elements of the sample space. Consider outcomes of Head–Tail and

Tail–Head both to be equivalent to an outcome of 1 Head and 1 Tail. Is each outcome equally likely?

3 Consider tossing a pair of coins 36 times. In theory, how many of each outcome should result?

Task 2Consider an experiment consisting of tossing two coins 36 times.

1 Each pair of participants should have two coins. Draw up the following table to record the tossing of the pair of coins 36 times.

When two coins are tossed there are three possible outcomes, 2 Heads, 2 Tails and one of each. Is each outcome equally likely?

THINK WRITE

There is more than one coin being tossed and so a tree diagram must be drawn.

There are actually four outcomes, two of which involve 1 Head and 1 Tail. Therefore each of the outcomes mentioned is not equally likely to occur.

Each outcome is not equally likely. There are two chances of getting one Head and one Tail. There is only one chance of getting 2 Heads and one chance of getting 2 Tails.

1

HeadHead

Tail

HeadTail

Tail

1st coin 2nd coin

2

13WORKEDExamplein

vestigation

investigation

Outcome Tally

HH

HT/TH

TT



2 Compare your tallies with the theoretical results you predicted in Task 1.

3 Compare your results with those obtained by other pairs in the class.

Task 3Set your calculator to generate random integers 1 or 2. Let a 1 represent a Tail and a 2 represent a Head.

1 With each person in the pair generating these random integers, simulate tossing two coins 36 times. Record your results in a table as you did in Task 2.

2 Compare these results with those obtained in Task 1 and Task 2.

3 Compare your answer with those from other pairs in the class.

Task 4This task uses a spreadsheet to simulate the tossing of the coins. Instructions and formulas relate to the Excel spreadsheet. If you are using a different one, your teacher will advise you of variations.

1 In the spreadsheet, type the entries shown in cells A1, E1, A3, D3, G3, E4, H4, E5, H5, E6, H6 and A8. Leave the cells I4, I5 and I6 blank at this stage. The results generated by the computer will be displayed there.

2 Enter the expected tally for each outcome in cells F4, F5 and F6.

3 The computer can randomly generate a 1 to represent a Tail or a 2 to represent a Head. To simulate the tossing of 2 coins, in the cell A10 enter the formula:

=IF(INT(RAND()*2+1)+INT(RAND()*2+1)=4,”HH”,IF(INT(RAND()*2+1)+INT)RAND()*2+1)=2,”TT”,”HT/TH”))

Copy this formula to the 36 cells in the range A10:I13.



4 To count the number of HH outcomes which result, in cell I4 enter the formula

=COUNTIF(A10:I13,”HH”)For the HT/TH count, in the cell I5 enter the formula

=COUNTIF(A10:I13,”HT/TH”)For the TT tally, in the cell I6, enter the formula

=COUNTIF(A10:I13,”TT”)

5 Simulate the 36 tosses by pressing the F9 key. Note the results. Repeat the simulation until you have accumulated the tallies of 10 experiments. Comment on the results.

Rolling two 6-sided diceResources: Pen, paper, two 6-sided dice, calculator, computer spreadsheet.

This activity is best performed in pairs and is designed as a follow-on from the investigation of tossing two coins. It relies on the skills developed previously, with only minimal guidance provided. (If you experience difficulties, consult the previous investigations where detailed information is supplied.) Approach this activity as you would an alternative assessment item, recording results in an orderly manner, culminating in the production of a report detailing your results.

Task 1Consider rolling two 6-sided dice 36 times. Explain the theoretical outcomes you would expect.

Task 2As a pair, roll two 6-sided dice 36 times. Record your results.

Task 3Use a calculator each to simulate the rolling of the two dice 36 times. Record your results.

Task 4Simulate the rolling of the two dice 36 times with the aid of a spreadsheet. Note your results.

ReportingWrite up the results of this investigation in the form of a report which must indicate detailed understanding of:a the theoretically expected resultsb the technique involved in undertaking the practical experiment and the results

obtainedc formulas used in the calculator simulation and the tallies resultingd formulas and results for the computer simulation — include at least two

printouts showing the diversity of resultse comparison of results from all tasksf summary and conclusion.

inve


nGC

program

Dice 2

EXCEL

Spreadsheet

Dice



Equally likely outcomes

1 A tennis match is to be held between Lindsay and Anna. There are two possible out-comes, Lindsay to win and Anna to win. Is each outcome equally likely? Explain youranswer.

2 There are 80 runners in the Olympic Games marathon. The sample space for thewinner of the race therefore has 80 elements. Is each outcome equally likely? Explainyour answer.

3 The numbers 1 to 40 are written on 40 marbles. The marbles are then placed in a bagand one is chosen from the bag. There are 40 elements to the sample space. Is eachoutcome equally likely? Explain your answer.

4 For each of the following, state whether each element of the sample space is equallylikely to occur.a A card is chosen from a standard deck.b The result of a volleyball game between two teams.c It will either rain or be dry on a summer’s day.d A raffle with 100 tickets has one ticket drawn to

win first prize.

5 For each of the following, state whether thestatement made is true or false. Give a reason foryour answer.a Twenty-six cards each have one letter of the

alphabet written on them. One card is thenchosen at random. Each letter of the alphabet hasan equal chance of being selected.

b A book is opened on any page and a pin is stuck inthe page. The letter closest to the pin is then noted. Eachletter of the alphabet has an equal chance of being selected.

6

In which of the following is each member of the sample space equally likely to occur?A Kylie’s softball team is playing a match that they could win, lose or draw.B A bag contains 4 red counters and 2 blue counters. One counter is selected from

the bag.C The temperature on a January day will be between 20°C and 42°C.D A rose is planted in the garden that may bloom to be red, yellow or white.

remember1. Each element of the sample space will not always be equally likely.2. Outcomes will be equally likely if a selection is random. When other factors

influence the selection, each outcome is not equally likely.3. When there is more than one event involved, examine the tree diagram to

determine if events described are equally likely.

remember

12DWORKEDExample

12




7 A couple have two children. They could have two boys, two girls or one of each. Thesample space therefore has three elements that are all equally likely. Is this statementcorrect? Explain your answer.

8 In a game, two dice are rolled and the total of the two dice is the player’s score.a What is the sample space for the totals of two dice?b Is each element of the sample space equally likely to occur?

9 A restaurant offers a three course meal. The menu is shown below.a A diner selects one plate from

each course. Draw a tree diagram to determine the number of elements in the sample space.

b Is each element of the sample space equally likely to occur?

10 There are 10 horses in a race. Ken hopes to select the winner of the race.a How many elements in the sample space?b Is each element of the sample space equally likely to occur? Explain your answer.c Loretta selects her horse by drawing the names out of a hat. In this case, is the

sample space the same? Is each element of the sample space equally likely tooccur? Explain your answer.

1 Describe the chance of selecting an Ace from a standard deck of cards.

For questions 2–5. A bag contains 3 black marbles, 4 white marbles and a red marble.

2 How many elements in the sample space?

3 How many distinct elements in the sample space?

4 If Julie needs to draw a red marble from the bag, how many favourable outcomes arethere?

5 Is each element of the sample space equally likely to occur?

6 A pair of twins is born. Draw a tree diagram and then list the sample space for allpossible combinations of boys and girls.

7 Amy and Samantha are in Year 11, while Luke, Matthew and John are in Year 12. OneYear 11 student and one Year 12 student are to represent the school at a conference.List the sample space for all pairs that could be chosen.

8 A two-digit number is formed using the digits 1, 2, and 3. How many elements has thesample space if the same digit can be used twice?

9 A two-digit number is formed using the digits 5, 6, and 7. How many elements in thesample space if the same digit cannot be used twice?

10 A student has an exam in mathematics that she could either pass or fail. Is eachelement of the sample space equally likely to occur?

WORKEDExample

13

Entree Main course

Prawn cocktailOystersSoup

Seafood platterChicken supremeRoast beefVegetarian quiche

Dessert

PavlovaIce-cream

2



Using the fundamental counting principle

A three-course meal is to be served at a 21st birthday party.Guests choose one plate from each course, as shown in themenu below.

In how many different ways can the three courses for the meal be chosen?

There are two possible choices of entree, four choices for main course and three dessert choices. To find the sample space for all possible outcomes, we draw a tree diagram.

By following the path to the end of each branch we can see that there are 24 elements in the sample space. If we simply need to know the number of elements in the sample space, we multiply the number of possible choices at each level.

Number of elements = 2 × 4 × 3= 24

There are 24 ways in which the three-course meal can be chosen.

This multiplication principle is called the fundamental counting principle.

The total number of ways that a succession of choices can be made is

found by multiplying the number of ways each single choice could be made.

The fundamental countingprinciple is used when eachchoice is made independently ofevery other choice. That is, whenone selection is made it has nobearing on the next selection. Inthe case above, the entree that ischosen has no bearing on whatmain course or dessert is chosen.

Entree Main course

Beef brothCalamari

SpaghettiRoast chickenPasta saladGrilled fish

Dessert

Ice-creamBanana splitStrawberries

Spaghetti

Beef broth

Roastchicken

Pasta salad

Grilled fish

Main courseEntree DessertIce-creamBanana splitStrawberries




Spaghetti

Calamari

Roastchicken

Pasta salad

Grilled fish







A poker machine has three wheels. There are 20 symbols on each wheel. In how many different ways can the wheels of the poker machine finish, once they have been spun?

THINK WRITE

There are 20 possibilities for how the first wheel can finish, 20 for the second wheel and 20 for the third wheel. Multiply each of these possibilities together.

Total outcomes = 20 × 20 × 20 = 8000

Give a written answer. There are 8000 different ways in which the wheels of the poker machine can land.

1

2

14WORKEDExample

In Year 11 at Blackhurst High School, there are four classes with 20, 22, 18 and 25 students in them respectively. A committee of four people is to be chosen, one from each class to represent Year 11 on the SRC. In how many ways can this group of four people be chosen?

THINK WRITE

There are 20 possible choices from the first class, 22 from the second, 18 from the third and 25 from the fourth class. Multiply these possibilities together.

Total possible outcomes = 20 × 22 × 18 × 25 = 198 000

Give a written answer. The committee of four people can be chosen in 198 000 different ways.

1

2

15WORKEDExample


C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 517Sometimes we need to reconsider examples that have some type of restriction placedon the possible selections.

If number plates consist of 3 letters and 3 digits, how many are possible if the first letter must be T, U or V, and the first digit cannot be 0 or 1?

THINK WRITE

There are 3 possible first letters.There are 26 possible second and third letters.There are 8 possible first digits.There are 10 possible second and third digits.Multiply all these possibilities together.

Total number plates = 3 × 26 × 26 × 8 × 10 × 10= 1 622 400

Give a written answer. There are 1 622 400 possible number plates under this system.

12

34

5

6

16WORKEDExample

remember1. The fundamental counting technique allows us to calculate the number of

different ways that separate events can occur.2. This method can be used only when each selection is made independently of

the others. To use this method, we multiply the number of ways that each selection can be made.

remember



Using the fundamental counting principle

1 A poker machine has four reels, with 15 symbols on each wheel. If the wheels arespun, in how many ways can they finish?

2 Consider each of the following events.a A 10c coin and a 20c coin are tossed.

In how many ways can they land?b A red die and blue die are cast.

How many ways can the two dice land?c A coin is tossed and a die is rolled.

How many possible outcomes are there?

3 A briefcase combination lock has a combination of three dials, each with 10 digits. How many possible combinations to the lock are there?

4 In the game of Yahtzee, five dice are rolled. In how many different ways can they land?

5 Some number plates have two letters followed by 4 numbers. How many of this style of plate are possible?

6

Personalised number plates that have six symbols can be any combination of letters ordigits. How many of these are possible?

7

A restaurant menu offers a choice of four entrees, six main courses and three desserts.If one extra choice is offered in each of the three courses, how many more combi-nations of meal are possible?

8 There are 86 students in Year 11 at Narratime High School. Of these, 47 are boys and39 are girls. One boy and one girl are to be chosen as school captains. In how manydifferent ways can the boy and girl school captain be chosen?

9 A travel agency offers Queensland holiday packages flying from Sydney withQANTAS and Ansett; travelling in First, Business and Economy class to Brisbane, theGold Coast, the Great Barrier Reef and Cairns for periods of 7, 10 and 14 days.How many holiday packages does the traveller have to choose from?

A 1 000 000 B 17 576 000 C 308 915 776 D 2 176 782 336

A 3 B 68 C 72 D 140

12EWORKEDExample

14



WORKEDExample

15


C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 51910 A punter at the racetrack tries to pick the Daily Double. This requires her to pick the

winner of race 6 and race 7. How many selections of two horses can she make if thereare:a 8 horses in each race?b 12 horses in each race?c 14 horses in race 6 and 12 in race 7?d 16 horses in race 6 and seven in race 7?e 24 horses in race 6 and 16 horses in race 7?

11 A poker machine has five wheels and 20 symbols on each wheel.a In how many ways can the wheels of the poker machine finish when spun?b There are 4 Aces on the first wheel, 5 on the second wheel, 2 on the third wheel, 6

on the fourth wheel and 1 on the fifth. In how many ways can five Aces be spun onthis machine?

12 Radio stations on the AM band have a call sign of a digit from 2 to 9, followed by twoletters.a How many radio stations could there be under this system?b In Qld all stations begin with a 4. How many stations are possible in Qld?

13 At a shoe store a certain pair of shoes can be bought in black, brown or grey; lace upor buckle up and in six different sizes. How many different pairs of shoes are poss-ible?

14 Home telephone numbers in Australia have eight digits.a How many possible home telephone numbers are there?b If a telephone number can’t begin with either a 0 or 1, how many are possible?c Freecall 1800 numbers begin with 1800 and then six more digits. How many of

these are possible?d A certain mobile network has numbers beginning with 015 or 018 followed by six

digits. How many numbers can this network have?

15 Madako can’t remember his PIN number for his bank account. He knows that it hasfour digits, does not begin with nine, is an odd number and that all digits are greaterthan five. How many possible PIN numbers could he try?

16

Postcodes in Australia begin with either 2, 3, 4, 5, 6, 7 or 8 followed by three moredigits. How many of these postcodes can there be?

17 Nadia goes to a restaurant that has a choice of 8 entrees, 15 main courses and 10 des-serts.a How many combinations of entree, main course and dessert are possible?b Nadia is allergic to garlic. When she examines the menu she finds that three

entrees and four main courses are seasoned with garlic. How many possiblechoices can she make without choosing a garlic dish?

18 Bill is trying to remember Tom’s telephone number. It has eight digits and Bill canremember that it starts with 963 and finishes with either a 4 or a 6. How many poss-ible telephone numbers could Tom try?

19 A representative from each of six classes must be chosen to go on a committee. Thereare four classes of 28 students, a class of 25 students and a class of 20 students. Howmany committees are possible?

A 70 B 1000 C 7000 D 10 000

WORKEDEExample

16

mmultiple choiceultiple choiceWo

rkSHEET 12.1



Q-lotto: FrequenciesNow that we have some understanding of the theory of chance, let us continue to investigate Sam’s chance of winning Q-lotto.

Each Q-lotto entry card has spaces to complete 12 games. Each game consists of 45 boxes. Six of these boxes are selected and marked with a cross.Consider the following questions:

1 Describe, in words, Sam’s chance of selecting one of the 6 correct numbers from the 45 available.

2 Describe Sam’s chance of selecting the correct 6 numbers from the 45 available.

3 Is he more or less likely to select the 6 correct than he is to choose 1 correct number?

4 Is it possible to select the same number more than once in each game?

5 List the sample space for each game. How many elements are in this sample space?

6 Considering the Q-lotto draw for Week 3 (see beginning of chapter), list the favourable outcomes.

7 How many elements are there in the set of favourable outcomes? Does this number change from week to week?

8 Of these elements, are they all required to win a prize? If not, how many are required?

9 Which ones are required to win the Division 1 prize?

10 Does each of the 45 numbers have an equal chance of being selected? Why/why not?

Let us consider the following statistics of frequencies of draws of the 45 balls in Q-lotto. (B1 = Ball no. 1; B2 = Ball no. 2 etc.; shaded rows show the number of times the ball number has been drawn.)

inve


n

1 2 3 4 5 6 7 8

9 10 11 12 13 14 15 16

17 18 19 20 21 22 23 24

25 26 27 28 29 30 31 32

33 34 35 36 37 38 39 40

41 42 43 44 45 Game 1

Number of times drawn

B1 B2 B3 B4 B5 B6 B7 B8 B9

60 54 54 65 48 43 60 55 70

B10 B11 B12 B13 B14 B15 B16 B17 B18

56 70 51 60 49 62 59 64 64

B19 B20 B21 B22 B23 B24 B25 B26 B27

55 62 60 52 52 59 50 58 71

B28 B29 B30 B31 B32 B33 B34 B35 B36

76 71 62 59 52 49 65 69 57

B37 B38 B39 B40 B41 B42 B43 B44 B45

56 70 65 60 55 59 70 50 55



521

Consider the following:

11

Do these tables reflect the numbers chosen by entrants?

12

How many times has the number 1 been drawn as one of the lucky numbers?

13

How many weeks is it since the number 1 was drawn?

14

What number has been drawn most frequently? How many times has it been drawn?

15

What number has been drawn least frequently? How many times has it been drawn?

16

Which numbers were drawn last week? You should have obtained eight numbers. Why are there 8 numbers and not only 6?

17

If you were to put in a Q-lotto entry next week, basing your numbers on those which have been chosen more than others, which numbers might you include in your 6?

18

Basing your Q-lotto entry next week on the fact that those numbers which haven’t turned up for a while might turn up next week, which numbers might you choose?

We’re well on the way to answering the lotto questions posed at the beginning of this chapter. We’ll resume our investigation after we consider relative frequency. At that stage, we should be able to verify Sam’s chance of picking the 6 winning numbers as quoted earlier in the chapter.

Number of weeks since drawn

B1 B2 B3 B4 B5 B6 B7 B8 B9

3 13 3 5 13 4 6 10 3

B10 B11 B12 B13 B14 B15 B16 B17 B18

0 4 18 1 1 13 8 1 6

B19 B20 B21 B22 B23 B24 B25 B26 B27

1 0 0 3 1 9 0 6 1

B28 B29 B30 B31 B32 B33 B34 B35 B36

1 8 9 1 6 8 2 2 5

B37 B38 B39 B40 B41 B42 B43 B44 B45

10 0 2 0 0 3 0 6 6


522


Relative frequency

You are planning to go skiing on the first weekend in July. The trip is costing you a lotof money and you don’t want your money wasted on a weekend without snow. So whatis the chance of it snowing on that weekend? We can use past records only to estimatethat chance.

If we know that it has snowed on the first weekend of July for 54 of the last 60 years,we could say that the chance of snow this year is very high. To measure that chance, wecalculate the

relative frequency

of snow on that weekend. We do this by dividing thenumber of times it has snowed by the number of years we have examined. In this case,we can say the relative frequency of snow on the first weekend in July is 54

÷

60

=

0.9.The relative frequency is usually expressed as a decimal and is calculated using the

formula:

Relative frequency

=

In this formula, a

trial

is the number of times the probability experiment has been conducted.

The relative frequency is used to assess the quality of products. This is done by findingthe relative frequency of defective products.

number of times an event has occurrednumber of trials

---------------------------------------------------------------------------------------------

The weather has been fine on Christmas Day in Sydney for 32 of the past 40 Christmas Days. Calculate the relative frequency of fine weather on Christmas Day.

THINK WRITE

Write the formula. Relative frequency =

Substitute the number of fine Christmas Days (32) and the number of trials (40).

Relative frequency =

Calculate the relative frequency as a decimal. = 0.8

1


---------------------------------------------------------------------------------------------

23240------

3

17WORKEDExample

A tyre company tests its tyres and finds that 144 out of a batch of 150 tyres will withstand 20 000 km of normal wear. Find the relative frequency of tyres that will last 20 000 km. Give the answer as a percentage.

THINK WRITE


Substitute 144 (the number of times the event occurred) and 150 (number of trials).


Calculate the relative frequency. = 0.96Convert the relative frequency to a percentage. = 96%

1


---------------------------------------------------------------------------------------------

2144150---------

34

18WORKEDExample



523

Relative frequencies can be used to solve many practical problems.

A batch of 200 light globes was tested. The batch is considered unsatisfactory if more than 15% of globes burn for less than 1000 hours. The results of the test are in the table below.

Determine if the batch is unsatisfactory.

No. of hours No. of globes

Less than 500 4

500 to less than 750 12


1000 to less than 1250 102


1500 or more 35

THINK WRITE

Count the number of light globes that burn for less than 1000 hours.

31 light globes burn for less than 1000 hours.


Substitute 31 (number of times the event occurs) and 200 (number of trials).


Calculate the relative frequency. = 0.155Convert the relative frequency to a percentage.

= 15.5%

Make a conclusion about the quality of the batch of light globes.

More than 15% of the light globes burn for less than 1000 hours and so the batch is unsatisfactory.

1

2


---------------------------------------------------------------------------------------------

331200---------

45

6

19WORKEDExample

remember1. The relative frequency is used to estimate the probability of an event.2. The relative frequency, usually expressed as a decimal, is a figure which

represents how often an event has occurred.3. The relative frequency is calculated using the formula:

Relative frequency = .

4. The relative frequency can also be written as a percentage and is used to solve practical problems.


---------------------------------------------------------------------------------------------

remember


524


Relative frequency

1

At the opening of the ski season, there has been sufficient snow for skiing for 37 outof the past 50 years. Calculate the relative frequency of sufficient snow at the begin-ning of the ski season.

2

A biased coin has been tossed 100 times with the result of 79 Heads. Calculate therelative frequency of the coin landing Heads.

3

Of eight maths tests done by a class during a year, Peter has topped the class threetimes. Calculate the relative frequency of Peter topping the class.

4

Farmer Jones has planted a wheat crop. For the wheat crop to be successful farmerJones needs 500 mm of rain to fall over the spring months. Past weather records showthat this has occurred on 27 of the past 60 years. Find the relative frequency of:

a

sufficient rainfall

b

insufficient rainfall.

5

Of 300 cars coming off an assembly line, 12 are found to have defective brakes.Calculate the relative frequency of a car having defective brakes. Give the answer as apercentage.

6

A survey of 25 000 new car buyers found that 750 cars had a major mechanicalproblem in the first year of operation. Calculate the relative frequency of the car:

a

having mechanical problems in the first year

b

not having mechanical problems in the first year.

7

On a production line, light globes are tested to see how long they will last. Aftertesting 1000 light globes, it is found that 960 will burn for more than 1500 hours.Wendy purchases a light globe. What is the relative frequency that the light globewill:

a

burn for more than 1500 hours?

b

burn for no more than 1500 hours?

8

A study of cricket players found that of 150 players, 36 batted left handed. What isthe relative frequency of left-handed batsmen?

9

Four surveys were conducted and the following results were obtained. Which resulthas the highest relative frequency?

A

Of 1500 P-plate drivers, 75 had been involved in an accident.

B

Of 1200 patients examined by a doctor, 48 had to be hospitalised.

C

Of 20 000 people at a football match, 950 were attending their first match.

D

Of 50 trucks inspected, 2 were found to be unroadworthy.

A

0.24

B

0.36

C 0.64 D 0.76

12FWORKEDExample

17

WORKEDExample

18




C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 52510 During an election campaign 2000 people were asked for their voting preferences.

One thousand and fifty said that they would vote for the government, 875 said theywould vote for the opposition and the remainder were undecided. What is the relativefrequency of:a government voters?b opposition voters?c undecided voters?

11 Research over the past 25 years shows that each November there is an average of twowet days on Sunnybank Island. Travelaround Tours offer one-day tours to SunnybankIsland at a cost of $150 each, with a money back guarantee against rain.a What is the relative frequency of wet November days as a percentage?b If Travelaround Tours take 1200 bookings for tours in November, how many

refunds would they expect to give?

12 An average of 200 robberies takes place each year in the town of Amiak. There are10 000 homes in this town.a What is the relative frequency of robberies in Amiak?b Each robbery results in an average insurance claim of $20 000. What would be the

minimum premium that the insurance company would need to charge to coverthese claims?

13 A car maker recorded the first time that its cars came in for mechanical repairs. Theresults are in the table below.

The assembly line will need to be upgraded if the relative frequency of cars needingmechanical repair in the first year is greater than 25%. Determine if this will be neces-sary.

14 For the table in question 13 determine, as a percentage, the relative frequency of:a a car needing mechanical repair in the first 3 monthsb a car needing mechanical repair in the first 2 yearsc a car not needing mechanical repair in the first 3 years.

Time taken No. of cars

0 to <3 months 5

3 to <6 months 12

6 to <12 months 37

1 to <2 years 49

2 to <3 years 62

3 or more years 35

WORKEDExample

19



15 A manufacturer of shock absorbers measures the distance that its shock absorbers cantravel before they must be replaced. The results are in the table below.

The relative frequency of the shock absorber lasting is 0.985 for a certain guaranteeddistance. What is the maximum distance the manufacturer will guarantee?

16 A soccer team plays 40 matches over a season and the results (wins, losses and draws)are shown below.

W W W D L L L D W L W D L DW W L L L D W W D L L W WW L D L D D L W W W D D L

a Put this information into a table showing the number of wins, losses and draws.b Calculate the relative frequency of each result over a season.

No. of kilometres No. of shock absorbers

0 to <20 000 1

20 000 to <40 000 2

40 000 to <60 000 46

60 000 to <80 000 61

80 000 to <100 000 90

Relative frequenciesResources: Calculator, World Wide Web.

Task 11 Set your calculator to generate random integers in the range 1 to 10 inclusive.

2 Draw up the table below to record the result of generating 100 random numbers:

3 Calculate the relative frequency of each number. Is it what you would have expected?

4 Repeat the experiment once more. Comment on any differences.

inve


n

Integer Tally FrequencyRelative

frequency

1

2

3

4

5

6

7

8

9

10



Single event probabilityPreviously we discussed the chances of certain events occurring. In doing so, we usedinformal terms such as probable and unlikely. However, while these terms give us anidea of whether something is likely to occur or not, they do not tell us how likely theyare. To do this, we need an accurate way of stating the probability.

We stated earlier that the chance of any event occurring was somewhere betweenimpossible and certain. We also said that:1. if an event is impossible the probability was 02, if an event is certain the probability was 1.It therefore follows that the probability of any event must lie between 0 and 1 inclusive.

A probability is a number that describes the chance of an event occurring. All prob-abilities are calculated as fractions but can be written as fractions, decimals or percent-ages. Probability is calculated using the formula:

The total number of favourable outcomes is the number of different ways the event canoccur, while the total number of outcomes is the number of elements in the samplespace.

Task 2The Bureau of Meteorology has a web site detailing the temperature, rainfall, cloudy days etc. for a large number of towns throughout Queensland. This is found at: http://www.bom.gov.au/climate

5 Visit the web site and search to locate the town closest to where you live. Choose some aspect of the climate and compile a relative frequency table displaying the occurrence and variation of the weather in your area. Write a conclusion highlighting your findings. Is it consistent with your experiences?

Task 3The World Wide Web records an abundance of statistical information — for example, results of sporting teams, stock market movements, Melbourne Cup winners, world leaders, movie attendance.

6 Choose a topic which interests you and research its statistics through the Web. Draw up a relative frequency table and summarise its results.

P event( ) number of favourable outcomestotal number of outcomes

----------------------------------------------------------------------------=

Zoran is rolling a die. To win a game, he must roll a number greater than 2. List the sample space and state the number of favourable outcomes.

THINK WRITE

There are 6 possible outcomes. S = {1, 2, 3, 4, 5, 6}The favourable outcomes are to roll a 3, 4, 5 or 6.

There are 4 favourable outcomes.12

20WORKEDExample



Consider the case of tossing a coin. If we are calculating the probability that it will landHeads, there is 1 favourable outcome out of a total of 2 possible outcomes. Hence wecan then write P(Heads) = . This method is used to calculate the probability of anysingle event.

In the above example the fraction could be simplified to .

Some questions have more than one favourable outcome. In these cases, we need to addtogether each of these outcomes to calculate the number of outcomes that are favourable.

12---

Andrea selects a card from a standard deck. Find the probability that she selects an Ace.

THINK WRITE

There are 52 cards in the deck (total number of outcomes).There are 4 Aces (number of favourable outcomes).Write the probability. P(Ace) =

1

2

3452------

21WORKEDExample

452------ 1

13------

In a barrel there are 6 red marbles, 2 green marbles and 4 yellow marbles. One marble is drawn at random from the barrel. Calculate the probability that the marble drawn is red.

THINK WRITE

There are 12 marbles in the barrel (total number of outcomes).There are 6 red marbles in the barrel (number of favourable outcomes).Write the probability. P(red) = =

1

2

3612------ 1

2---

22WORKEDExample

On a bookshelf there are 4 history books, 7 novels, 2 dictionaries and 5 sporting books. If I select one at random, what is the probability that the one chosen is not a novel?

THINK WRITE

There are 20 books on the shelf (total number of outcomes).Seven of these books are novels, meaning that 13 of them are not novels (number of favourable outcomes).Write the probability. P(not a novel) =

1

2

31320------

23WORKEDExample


C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 529Some questions do not require us to calculate the entire sample space, only the samplespace for a small part of the experiment.

The digits 1, 3, 4, 5 are written on cards and these cards are then used to form a four-digit number. Calculate the probability that the number formed is:a evenb greater than 3000.

THINK WRITE

a If the number is even the last digit must be even.

a

There are four cards that could go in the final place (total number of outcomes).Only one of these cards (the 4) is even (number of favourable outcomes).

Write the probability. P(even) =

b If the number is greater than 3000, then the first digit must be a 3 or greater.

b

There are four cards that could go in the first place.Three of these cards are a 3 or greater.

Write the probability. P(greater than 3000) =

1

2

3

414---

1

2

3

434---

24WORKEDExample

remember1. The sample space is the list of all possible outcomes in a probability

experiment.2. The probability of an event is calculated using the formula:

P(event) = number of favourable outcomes

total number of outcomes----------------------------------------------------------------------------

remember



Single event probability

1 A coin is tossed at the start of a cricket match. Manuel calls Heads. List the samplespace and the number of favourable outcomes.

2 For each of the following probability experiments, list the sample space and state the number of favourable outcomes.

a Rolling a die and needing a 6

b Rolling two dice and needing a total greater than 9

c Choosing a letter of the alphabet and it being a vowel

d The chance a baby will be born on the weekend

e The chance that a person’s birthday will fall in summer

3 For each of the following probability experiments, state the number of favourableoutcomes and the total number of outcomes.

a Choosing a red card from a standard deck

b Selecting the winner of a 15-horse race

c Selecting the first ball drawn in a lotto draw (The balls are numbered 1 to 45.)

d Winning a raffle with 5 tickets out of 1500

e Selecting a yellow ball from a bag containing 3 yellow, 4 red and 4 blue balls

4 A coin is tossed. Find the probability that the coin will show Tails.

5 A regular die is rolled. Calculate the probability that the uppermost face is:

6 A barrel contains marbles with the numbers 1 to 45 written on them. One marble isdrawn at random from the bag. Find the probability that the marble drawn is:

a 6 b 1 c an even number

d a prime number e less than 5 f at least 5.

a 23 b 7 c an even number

d an odd number e a multiple of 5 f a multiple of 3

g a number less than 20 h a number greater than 35 i a square number.

12GWORKEDExample

20

WORKEDExample

21



531

7

Many probability questions are asked about decks of cards. You should know thecards making up a standard deck.

A card is chosen from a standard deck. Find the probability that the card chosen is:

8

A bag contains 12 counters: 7 are orange, 4 are red and 1 is yellow. One counter isselected at random from the bag. Find the probability that the counter chosen is:

9

The digits 2, 3, 5 and 9 are written on cards. One card is then chosen at random. Findthe probability that the card chosen is:

10

In a bag of fruit there are 4 apples, 6 oranges and 2 pears. Larry chooses a piece offruit from the bag at random but he does not like pears. Find the probability that Larrydoes not select a pear.

11

The digits 2, 3, 5 and 9 are written on cards. They are then used to form a four-digitnumber. Find the probability that the number formed is:

12

A die is rolled. The probability that the number on the uppermost face is less than 4 is:

a

the Ace of diamonds

b

a King

c

a club

d

red

e

a picture card

f

a court card.

a

yellow

b

red

c

orange.

a

the number 2

b

the number 5

c

even

d

odd

e

divisible by 3

f

a prime number.

a

even

b

odd

c

divisible by 5

d

less than 3000

e

greater than 5000.

A B C D

KQ

J 10 9 8 7 6 5 4 3 2

2

A

KQ

J 10 9 8 7 6 5 4 3 2

A

2

KQ

J 10 9 8 7 6 5 4 3 2

A

2

KQ

J 10 9 8 7 6 5 4 3 2

A

2

WORKEDExample

22

WORKEDExample

23

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24


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MQ Maths A Yr 11 - 12 Page 531 Thursday, July 5, 2001 12:09 PM


13

When a die is rolled, which of the following outcomes does not have a probabilityequal to ?A The number on the uppermost face is greater than 3.B The number on the uppermost face is even.C The number on the uppermost face is at least a 3.D The number on the uppermost face is a prime number.

14

A card is chosen from a standard deck. The probability that the card chosen is a courtcard is:

15

When a card is chosen from a standard deck, which of the following events is mostlikely to occur?

16 One thousand tickets are sold in a raffle. Craig buys five tickets.a One ticket is drawn at random. The holder of that ticket wins first prize. Find the

probability of Craig winning first prize.b After the first prize has been drawn, a second prize is drawn. If Craig won first

prize, what is the probability that he now also wins second prize?

17 A lottery has 160 000 tickets. Janice buys one ticket. There are 3384 cash prizes in thelottery.a What is the probability of Janice winning a cash prize?b If there are 6768 consolation prizes of a free ticket for being one number off a

cash prize, what is the probability that Janice wins a consolation prize?c What is the probability that Janice wins either a cash prize or a consolation prize?

18 A number is formed using all five of the digits 1, 3, 5, 7 and 8. What is the probabilitythat the number formed:

19 Write down an example of an event which has a probability of:

20 A three-digit number is formed using the digits 2, 4 and 7.a Explain why it is more likely that an even number will be formed than an odd

number.b Which is more likely to be formed: a number less than 400 or a number greater

than 400?

A B C D

A choosing a seven B choosing a clubC choosing a picture card D choosing a black card

a begins with the digit 3 b is even c is oddd is divisible by 5 e is greater than 30 000 f is less than 20 000.

a b c .


12---


152------ 1

13------ 3

13------ 4

13------


12--- 1

4--- 2

5---



533

1

A die is rolled. Find the probability that the uppermost face is 4.

2

A card is drawn from a standard pack. Find the probability of selecting a Jack.

3

A bag contains four $1 coins and seven $2 coins. Find the probability that a coindrawn at random from the bag will be a $2 coin.

4

A barrel, containing balls numbered 1 to 100, has one ball selected at random fromit. Find the probability that the ball selected is a multiple of 3.

5

Five history books, 3 reference books and 10 sporting books are arranged on ashelf. What is the probability of a sporting book being on the left-hand end of thebookshelf?

6

A coin is tossed 10 times with a result of 7 Heads and 3 Tails. The relativefrequency of this coin landing Heads is 0.7; true or false?

7

A coin is tossed 10 times with a result of 7 Heads and 3 Tails. The probability ofthis coin landing Tails is ; true or false?

8

In 60 rolls of a die, there have been 12 sixes. What is the relative frequency ofrolling a six?

9

During a football season a team has won 15 matches and lost 5. Calculate therelative frequency of the team winning.

10

A car assembly line finds that five in every 1000 cars have faulty paintwork. If Ipurchase one of these cars find, as a percentage, the relative frequency that thepaintwork is faulty.

3

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MQ Maths A Yr 11 - 12 Page 533 Thursday, July 5, 2001 12:09 PM


Comparing probabilities with actual results

In this activity, we compare the probability of certain events to practical results. You may be able to do a simulation of these activities on a spreadsheet.

Tossing a coinIf we toss a coin, P(Heads) = .1 If you toss a coin, how many Heads would you expect in:

a 4 tosses b 10 tosses c 50 tosses d 100 tosses.2 Toss a coin 100 times and record the number of Heads after:

a 4 tosses b 10 tosses c 50 tosses d 100 tosses.3 Combine your results with those of the rest of the class. How close to 50% is

the total number of Heads thrown by the class?

Rolling a die1 When you roll a die, what is the probability of rolling a 1? (In fact, the

probability for each number on the die is the same.)2 Roll a die 120 times and record each result in the table below.

How close are the results to the results that were expected?

Rolling two dice1 Roll two dice and record the total on the faces of the two dice. Repeat this 100

times and complete the table below.

2 Do you notice anything different about the results of this activity, compared to the others?

inve


n

12---

Number Occurrences Percentage of throws123456

Number Occurrences Percentage of throws23456789

101112



Writing probabilities as decimals and percentages

In our exercises so far, we have been writing probabilities as fractions. This is the waythat most mathematicians like to express chance. However, in day-to-day language,decimals and percentages are also used. Therefore, we need to be able to write prob-abilities as both decimals and percentages.

When writing a probability as a decimal, we use the same formula and divide thenumerator by the denominator to convert to a decimal.

The chance of an event occurring is commonly expressed as a percentage. This is thepercentage chance of an event occurring. When writing a probability as a percentage,we take the fractional answer and multiply by 100% to convert to a percentage.

If I select a card from a standard deck, what is the probability of selecting a heart, expressed as a decimal?

THINK WRITE

There are a total of 52 cards in the deck (elements of the sample space).There are 13 hearts in the deck (elements of the event space).Write the probability. P(heart) =

Convert to a decimal. = 0.25

1

2

31352------

4

25WORKEDExample

In a bag there are 20 counters: 7 are green, 4 are blue and the rest are yellow. If I select one at random, find the probability (as a percentage) that the counter is yellow.

THINK WRITE

There are 20 counters in the bag (elements of the sample space).There are 9 yellow counters in the bag (elements of the event space).Write the probability. P(yellow counter) = × 100%

Convert to a percentage. = 45%

1

2

3920------

4

26WORKEDExample

remember1. Sometimes it is necessary to write a probability as a decimal or a percentage.2. To write a probability as a decimal, we calculate the probability as a fraction,

then divide the numerator by the denominator to convert to a decimal.3. To write a probability as a percentage, we calculate the probability as a

fraction, then multiply by 100% to convert to a percentage.

remember



Writing probabilities as decimals and percentages

1 A die is rolled. What is the probability of rolling an even number, expressed as adecimal?

2 A barrel contains 40 marbles. There are 10 blue marbles, 15 red marbles and 15 white marbles. A marble is selected at random from the barrel. Calculate as a percentage the probability of selecting a red marble.

3 Write down the probability that a tossed coin will land Tails:a as a decimalb as a percentage.

4 A student is rolling a die. Write down each of the following probabilities as decimals, correct to 2 decimal places.a Getting a 1b Getting an odd numberc Getting a number greater than 4

5 For rolling a die, write down the following probabilities as percentages. Give youranswers correct to 1 decimal place.a Getting a 3b Getting an even numberc Getting a number less than 6

6 From a standard deck of cards, one is selected at random. Write down the probabilityof each of the following as a decimal (correct to 2 decimal places where necessary).a Selecting the King of heartsb Selecting a spadec Selecting any 5d Selecting a red carde Selecting a court card (any King, Queen or Jack — the Jack is also called a

Knave)

7 When selecting a card from a standard deck, what would be the probabilities of thefollowing, written as percentages? Give your answers correct to 1 decimal place.a Selecting a Jack of clubsb Selecting a diamondc Selecting any 2d Selecting a black carde Selecting a court card

12H

SkillSH

EET 12.2WORKEDExample

25

SkillSH

EET 12.3

WORKEDExample

26



537

8

A raffle has 400 tickets. Sonya has bought 8 tickets. The probability that Sonya winsfirst prize in the raffle is:

9

In a class of 25 students, there are 15 boys and 10 girls. If a student is chosen atrandom from the class, the probability that the student is a boy is:

10

Which of the following does

not

describe the chance of selecting a diamond from astandard deck of cards?

11

The diagram on the right shows a spinner that can be used in a board game. When the player spins the spinner, what is the probability of getting the following results (expressed as a decimal)?

a

A 5

b

An even number

c

An odd number

d

A number greater than one

12

The board game in question

11

has the following rules. A player spinnning a 2 or a 5is out of the game. A player spinning a 3 collects a treasure and automatically winsthe game. Write down the probability, as a percentage, that with the next spin aplayer:

a

wins the game

b

is out of the game

c

neither wins nor is out of the game.

A

0.02

B

0.08

C

0.2

D

0.8

A

10%

B

15%

C

40%

D

60%

A B

0.13

C

0.25

D

25%




1352------

1

23

4

5


538


Range of probabilities

Consider the following problem:A die is rolled. Calculate the probability that the uppermost face is a number less than 7.

We know this is certain to occur but we will look at the solution using the probabilityformula. There are 6 elements in the sample space and 6 elements that are favourable.Therefore:

P(no. less than 7)

=

=

1

When the probability of an event is 1, the event is certain to occur.

Now let’s consider an impossible situation:A die is rolled. Calculate the probability that the uppermost face is a number greaterthan 7.

There are 6 elements in the sample space and there are 0 elements that are favour-able. Therefore:

P(no. greater than 7)

=

=

0

When the probability of an event is 0, the event is impossible.

All probabilities therefore lie in the range 0 to 1. An event with aprobability of has an even chance of occurring or not occurring.

The range of probabilities can be seen in the figure at right.This figure allows us to make a connection between the formal

probabilities that we calculated in the previous exercise, and the informal terms we used earlier in the chapter.

The closer a probability is to 0, the less likely it is to occur. The closer the probability is to 1, the more likely it is to occur.

0

≤

P(E)

≤

1We read this as: ‘The probability of an event is greater than or equal to zero, and lessthan or equal to 1’.

66---

06---

Very unlikelyImpossible

Unlikely

Fifty-fifty

Probable

CertainAlmost certain

1–2

1

0

12---

For the following probabilities, describe whether the event would be certain, probable, fifty-fifty, unlikely or impossible.a b 0 c

THINK WRITE

a is less than and is therefore unlikely to occur.

a The event is unlikely as it has a probability of less than .

b A probability of 0 means the event is impossible.

b The event is impossible as it has a probability of 0.

c = . Therefore, the event has an even chance of occurring.

c The event has an even chance of occurring as the probability = .

49--- 18

36------

49--- 1

2---

12---

1836------ 1

2---

12---

27WORKEDExample



539

You need to be able to recognise when you can and cannot measure the probability. Youcannot measure probability when each outcome is not equally likely.

In a batch of 400 televisions, 20 are defective. If one television is chosen, find the probability of its not being defective and describe this chance in words.

THINK WRITE

There are 400 televisions (elements of the sample space).There are 380 televisions which are not defective (number of favourable outcomes).Write the probability. P(not defective) =

=

Since the probability is much greater than and very close to 1, it is very probable that it will not be defective.

It is very probable that the television chosen will not be defective.

1

2

3380400---------

1920------

412---

28WORKEDExample

State whether the following statements are true or false, and give a reason for your answer.a The probability of correctly selecting a number between 1 and 10 drawn out of a barrel

is .b The weather tomorrow could be fine or rainy, therefore the probability of rain is .

THINK WRITE

a Each outcome is equally likely. a True, because each number is equally likely to be selected.

b Each outcome is not equally likely. b False, because there is not an equal chance of the weather being fine or rainy.

110------

12---

29WORKEDExample

remember1. Probabilities range from 0 to 1. A probability of 0 means that the event is

impossible, while a probability of 1 means the event is certain.2. By calculating the probability, we are able to make a connection with the more

informal descriptions of chance.3. The rules of probability can be applied only when each outcome is equally

likely to occur.

remember



Range of probabilities

1 For each of the probabilities given below, state whether the event would be imposs-ible, unlikely, even chance, probable or certain.

2 For each of the events below, calculate the probability and hence state whether theevent is impossible, unlikely, even chance, probable or certain.a Rolling a die and getting a negative numberb Rolling a die and getting a positive numberc Rolling a die and getting an even numberd Selecting a card from a standard deck and getting a red carde Selecting a card from a standard deck and getting an Acef Reaching into a moneybox and selecting a 30c pieceg Selecting a blue marble from a bag containing 3 red, 3 green and 6 blue marbles

3 Give an example of an event with a probability which is:

4 The probabilities of five events are given below. Write these in order from the mostlikely to the least likely event.

5 By calculating the probability of each, write the following events in order from leastto most likely.A — Winning a raffle with 5 tickets out of 30B — Rolling a die and getting a number less than 3C — Drawing a green marble from a bag containing 4 red, 5 green and 7 blue marblesD — Selecting a court card from a standard deckE — Tossing a coin and having it land Heads

6

The probabilities of several events are shown below. Which of these is the most likelyto occur?

7

Cards in a stack have the letters of the alphabet written on them (one letter per card).Vesna draws a card from the stack. The probability of selecting a card that has a vowelwritten on it could best be described as:

a b c

d 1 e f

g h 0 i

a certain b probable c even chanced unlikely e impossible.

A B C D

A impossible B unlikely C even chance D probable

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For which of the following events can the probability not be calculated?A Selecting the first number drawn from

a barrel containing 20 numbered marblesB Selecting a diamond from a standard

deck of cardsC Winning the lottery with one ticket

out of 150 000D Selecting the winner of the Olympic

100-metre final with 8 runners

9 In a batch of 2000 cars that come off an assembly line, 50 have faulty paintwork. A car is chosen at random.a Find the probability that it has faulty

paintwork.b Describe the chance of buying a car with

faulty paintwork.

10 A box of matches has on the label ‘Minimum contents 50 matches’. The qualitycontrol department of the match manufacturer surveys boxes and finds that 2% ofboxes have less than 50 matches. Find the probability of a box containing at least 50matches and hence describe the chance that the box will contain at least 50 matches.

11 A box of breakfast cereal contains a card on which there may be a prize. In every100 000 boxes of cereal the prizes are:

1 new car5 Disneyland holidays50 computers2000 prizes of $100 in cash50 000 free boxes of cereal

All other boxes have a cardlabelled ‘Second Chance Draw’.Describe the chance of getting a cardlabelled:a new carb free box of cerealc any prized ‘Second Chance Draw’.

12 For each of the following, determine whether the statement is true or false, giving areason for your answer.a The probability of selecting an Ace from a standard deck of cards is .b The probability of selecting the letter P from a page of a book is .c In a class of 30 students, the probability that Sam tops the class in a maths test is

.

d In a class of 30 students the probability that Sharon’s name is drawn from a hat is

.


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1 A coin is tossed. Find the probability that the coin will land Heads.

2 A card is drawn from a standard deck. Find the probability that the card selected is adiamond.

3 Three events have probabilities , and . List these from the least likely to the most likely.

4 A tennis club has 40 members, of which 25 are female. If one member is chosen at random, find the probability (as a percentage) that the member is female.

5 For the tennis club in question 4, what is the probability (as a decimal) that the member chosen is male?

6 A card is drawn from a standard deck. Find the probability that the card selected is either a King or a Queen (as a decimal to 3 decimal places).

7 A card is drawn from a standard deck. Find the probability that the card selected is a picture card (as a percentage to 1 decimal place).

Copy and complete questions 8–10.

8 If an event is certain then the probability of its occurring is .

9 If an event is impossible then the probability of its occurring is .

10 An event has a probability of . The likelihood that the event will occur could be described as .

4

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Complementary eventsWhen tossing a coin, we know there are two elements in the sample space:

P(Heads) = and P(Tails) =

The total of the probabilities is 1. Now consider a slightly more difficult problem.

In any probability experiment the total of all probabilities equals 1.

Graphing resultsWeather statistics1 Use the Internet to find the number of wet days in Brisbane during each month

of the last five years. Copy and complete the table below for each month of the year.

2 Set up a spreadsheet to display the date.

3 Graph the month against the relative frequency of rain.

Sporting resultsChoose a sporting competition such as the AFL or NRL. 1 Use the current or most recent season to calculate the relative frequency of each

team’s winning.

2 Choose an appropriate graph to display the results.(If you are using a spreadsheet, you can easily update your results each week.)

Topic of interestChoose a topic of interest. Research your area thoroughly and display your findings in graphical form.

Year No. of wet days Relative frequency

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In a bag with 10 counters, there are 7 black and 3 white counters. If one counter is selected at random from the bag, calculate:a the probability of selecting a white counterb the probability of selecting a black counterc the total of the probabilities.

THINK WRITE

a There are 10 counters of which 3 are white. a P(white) =

b There are 10 counters of which 7 are black. b P(black) =

c Add and together. c Total = + = 1

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10------ 3

10------ 7

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We can use this rule to help us make calculations. In the above example, the chanceof selecting a black counter and the chance of selecting a white counter are said to becomplementary events. Complementary events are two events for which the prob-abilities have a total of 1. In other words, complementary events cover all possible out-comes to the probability experiment.

When we are given one event and asked to state the complementary event, we needto describe what must happen for the first event not to occur.

We can use our knowledge of complementary events to simplify the solution to manyproblems. The probability of an event and its complement will always add to give 1.We can use the result:

P(an event does not occur) = 1 − P(the event does occur)

For each of the following events, write down the complementary event.

a Tossing a coin and getting a Headb Rolling a die and getting a number less than 5c Selecting a heart from a standard deck of cards

THINK WRITE

a There are two elements to the sample space, Heads and Tails. If the coin does not land Heads, it must land Tails.

a The complementary event is that the coin lands Tails.

b There are 6 elements to the sample space — 1, 2, 3, 4, 5, and 6. If we do not get a number less than 5 we must get either a 5 or a 6.

b The complementary event is that we get a number not less than 5; that is, 5 or more.

c As we are concerned with only the suit of thecard, there are four elements to the samplespace: hearts, diamonds, clubs and spades. Ifwe do not get a heart we can get any other suit.

c The complementary event is that we do not get a heart; that is, we get a diamond, club or spade.

31WORKEDExample

Jessie has a collection of 50 CDs. Of these, 20 are by a rap artist, 10 are by heavy metal performers and 20 are dance music. If we select one CD at random, what is the probability that it is:a a heavy metal CD b not a heavy metal CD.

THINK WRITE

a Of 50 CDs, 10 are by heavy metal performers.

a P(heavy metal CD) =

=

b This is the complement of selecting a heavy metal CD. Subtract the probability of selecting a heavy metal CD from 1.

b P(not heavy metal) = 1 − P(heavy metal)= 1 − =

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Complementary events

1 A die is rolled.a List the sample space.b Write down the probability of each event in the sample space.c What is the total of the probabilities?

2 A barrel contains 20 marbles. We know that 7 of them are blue, 8 are red and the restare yellow.a One marble is selected from the barrel. Calculate the probability that it is:

i blue ii red iii yellow.b Calculate the total of these probabilities.

3 For each of the following, state the complementary event.a Winning a raceb Passing a testc Your birthday falling on a Monday

4 Match each event in the left-hand column with the complementary event in the right-hand column.A coin landing Heads A coin landing TailsAn odd number on a die A non-picture card from a standard deckA picture card from a standard deck Not winning 1st prize in the raffleA red card from a standard deck A team not making the last four Winning 1st prize in a raffle An even number on a diewith 100 tickets A black card from a standard deckMaking the last 4 teams in a 20-team tournament

5 For each pair of events in question 4, calculate:a the probability of the event in the left-hand columnb the probability of its complementary eventc the total of the probabilities.

6 You are rolling a die. Write down the complementary event to each of the following.a Rolling an even numberb Rolling a number greater than 3c Rolling a number less than 3d Rolling a 6e Rolling a number greater than 1

remember1. The complement of an event is the event that describes all other possible

outcomes to the probability experiment.2. The sum of the probability of an event and its complement equals 1.3. To calculate the probability of an event, subtract the probability of its

complementary event from 1.

remember

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7 In a barrel there are balls numbered 1 to 45. For each of the following, write down thecomplementary event.a Choosing an odd-numbered ballb Choosing a ball numbered less than 20c Choosing a ball that has a number greater than 23d Choosing a ball that is a multiple of 5

8 In a barrel there are 25 balls, 15 of which are coloured (10 pink and 5 orange). Therest are black. What is the complementary event to selecting:a a black ball?b a coloured ball?c a pink ball?

9

Wilson rolls two dice. He needs to get a 6 on at least one of the dice. What is the com-plementary event?A Rolling no sixesB Rolling 2 sixesC Rolling 1 sixD Rolling at least 1

10

The probability of rolling at least one six is . What is the probability of the comple-mentary event?

11 In a barrel with 40 marbles, 20 are yellow, 15 are green and 5 are orange. If onemarble is selected from the bag find the probability that it is:a orangeb not orange.

12 In a barrel there are 40 balls numbered 1 to 40. One ball is chosen at random from thebarrel.a Find the probability that the number is a multiple of 5.b Use your knowledge of complementary events to find the probability that the

number is not a multiple of 5.

13 There are 40 CDs in a collection. They can be classified as follows.

18 heavy metal6 rock10 techno6 classical

If one CD is chosen at random, calculate the probability that it is:a heavy metalb not heavy metalc classicald not classicale heavy metal or rockf techno or classical.

A B C D 1


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C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 54714 In a golf tournament there are 40 players. Of these, 16 are Australian and 12 are

American. If they are all of the same skill level, find the probability that the tourna-ment is:

15 After studying a set of traffic lights, Karen found that in every 100 seconds they werered for 60 seconds, amber for 5 seconds and green for 35 seconds. If you were toapproach this set of lights calculate the probability that:a they will be green b you will need to stop.

16 In a game of Scrabble there are 100 lettered tiles. These tiles include 9 ‘A’s, 12 ‘E’s,9 ‘I’s, 8 ‘O’s and 4 ‘U’s. One tile is chosen. Find the probability that it is:a an ‘E’ b a vowel c a consonant.

17 From past performances it is known that a golfer has a probability of 0.7 of sinking aputt. What is the probability that he misses the putt?

18 A basketballer is about to take a shot from the free throw line. His past record showsthat he has a 91% success rate from the free throw line. What would be the relativefrequency (as a percentage) of his:a being successful with the shot? b missing the shot?

1 A card is drawn from a standard deck and its suit noted. List the sample space forthis experiment.

2 Andrew needs to ring Sandra but he has forgotten the last digit. Find the probabilitythat he can correctly guess the number.

3 If Andrew knows that the last digit of a telephone number is not a 7 or a 0, what isthe probability of guessing the number?

4 What is the probability of correctly guessing the 4-digit PIN number to a bankaccount card?

5 A bead is selected from a bag containing 3 red beads, 4 yellow beads and 8 bluebeads. Find the probability that the bead selected is blue.

6 What is the probability that the bead selected in question 5 is not blue?

7 A number is chosen in the range 1 to 20. Find the probability the number chosen isa multiple of 3.

8 A number is chosen in the range 1 to 20. Find the probability the number chosen isa multiple of 5.

9 Find the probability that the number chosen is not a multiple of 5.

10 Find the probability that the number chosen is not a square number.

a won by an Australian b won by an Americanc not won by an Australian d not won by an Americane not won by an Australian or an American.

WorkS

HEET 12.2

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Q-lotto: ConclusionWe’re now in a position to conclude our investigation into Sam’s chance of winning Q-lotto.

In order to win the jackpot, Sam needs to choose 6 correct numbers. For his first selection, he has a choice of 45 numbers. Because he can’t select the same number for his second choice, he has only 44 numbers to choose from. By the same reasoning, he has a choice out of 43 numbers for his third, 42 for his fourth and so on.

1 How many choices does Sam have in selecting his 6 numbers from the 45 available?

2 Let’s say the six correct numbers are

1 2 3 4 5 6.

The following choice of numbers would be just as correct

2 1 3 4 5 6

as would the choice

3 2 1 4 5 6

and other combinations of the six numbers. It is obvious that all combinations of these six numbers would constitute winning entries.How many combinations of the six numbers are possible?

3 So, taking into account the fact that your answer in part 1 included all these alternative combinations of the correct six numbers, how many ways can six numbers be chosen from 45 when order is not important?

4 Of all these choices, only one constitutes the correct six numbers. So, what is the probability that these are the correct six? Did your answer agree with the figure quoted at the beginning of the chapter?

The calculations involved in verifying the probabilities of winning lower division prizes are quite complex, and beyond the scope of this course. Suffice to say that Sam has very little chance of becoming a millionaire by submitting Q-lotto entries each week.

inve


n



Informal description of chance• The chance of an event occurring can be described as being from certain (a

probability of 1) to impossible (a probability of 0).• Terms used to describe the chance of an event occurring include improbable,

unlikely, fifty-fifty, likely and probable.• The chance of an event occurring can be described by counting the possible

outcomes and sometimes by relying on our general knowledge.

Sample space• Sample space is a list of all possible outcomes to a probability experiment.• It includes every possible outcome even if some outcomes are the same.

Tree diagrams• Tree diagrams are used to list the sample space when there is more than one stage

to a probability experiment.• The tree must branch out once for each stage of the probability experiment.

Equally likely outcomes• Equally likely events occur when the selection method is random.• Events will not be equally likely when other factors influence selection. For

example, in a race, each person will not have an equal chance of winning, as each runner will be of different ability.

The fundamental counting principle• This principle can be used to count the number of elements in a sample space of a

multi-stage experiment.• The total number of possible outcomes is calculated by multiplying the number of

ways each stage of the experiment can occur.

Relative frequency• Relative frequency describes how often an event has occurred.• It is found by dividing the number of times an event has occurred by the total

number of trials.

Single event probability• The probability of an event can be found using the formula:

P(event) =

• Probabilities are usually written as fractions but can also be expressed as decimals or percentages.

Range of probabilitiesProbabilities range from 0 (impossible) to 1 (certain). The use of a fraction for a probability can help us describe, in words, the chance of an event occurring.

summary

number of favourable outcomestotal number of outcomes

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Complementary events

• The complement of an event is the event that describes all other possible outcomes to the probability experiment.

• The probability of an event and its complement add to give 1.• The probability of an event can often be calculated by subtracting the probability of

its complementary event from 1.

MQ Maths A Yr 11 - 12 Page 550 Friday, July 6, 2001 2:53 PM


1 Graham and Marcia are playing a game. To see who starts they each take a card from a standard deck. The player with the highest card starts. Graham takes a five. Describe Marcia’s chance of taking a higher card.

2 Describe each of the following events as being either certain, probable, even chance (fifty-fifty), unlikely or impossible.

a Rolling a die and getting a number less than 6

b Choosing the eleven of diamonds from a standard deck of cards

c Tossing a coin and having it land Tails

d Rolling two dice and getting a total of 12

e Winning the lottery with one ticket

3 Give an example of an event which is:

a certain b impossible.

4 The Chen family are going on holidays from Tasmania to Queensland during January. Are they more likely to experience hot weather or cold weather?

5 List each of the events below in order from most likely to least likely.Winning a lottery with 1 ticket out of 100 000 tickets soldRolling a die and getting a number greater than 1Selecting a blue marble out of a bag containing 14 blue, 15 red and 21 green marblesSelecting a picture card from a standard deck

6 Mark and Lleyton are tennis players who have played eight previous matches. Mark has won six of these matches. When they play their ninth match, who is more likely to win? Explain your answer.

7 The letters of the word SAMPLE are written on cards and placed face down. One card is then selected at random. List the sample space.

8 List the sample space for each of the following probability experiments.

a A coin is tossed.

b A number is selected from the numbers 1 to 18.

c The four Aces from a deck of cards are selected. One of these cards is then chosen.

d A bag contains 4 black marbles, 3 white marbles and 5 green marbles. One marble is then selected from the bag.

9 To win a game, Sarah must roll a number greater than 4 on the die.

a List the sample space.

b List the favourable outcomes.

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10 For each of the following, state:i the number of elements in the sample spaceii the number of favourable outcomes.

a At the start of a cricket match, a coin is tossed and Steve calls Heads.b Anne selects a card from a standard deck and needs a number less than 9. (Aces count as 1.)c A bag contains 3 red, 8 blue and 4 black discs. Florian draws a disc from the bag and

must not draw a black disc.

11 Two coins are tossed. Draw a tree diagram to find the sample space.

12 Two dice are rolled. How many elements are in the sample space?

13 A two-digit number is formed using 5, 6, 7 and 9, without repetition.a Use a tree diagram to list the sample space.b If Dan wants to make a number greater than 60, how many favourable outcomes are

there?

14 Mary, Neville, Paul, Rachel and Simon are candidates for an election. There are two positions, president and vice-president. One person cannot hold both positions.a List the sample space.b If Paul is to hold one of the positions, how many elements has the event space?

15 A school must elect one representative from each of three classes to sit on a committee. In 11A the candidates are Tran and Karen. In 11B the candidates are Cara, Daisy, Henry and Ian. In 11C the candidates are Bojan, Melina and Zelko.a List the sample space.b If there is to be at least one boy and at least one girl on the committee, how many

elements are in the sample space?

16 A greyhound race has eight runners.a How many elements has the sample space?b Is each element of the sample space equally likely to occur? Explain your answer.

17 For each of the following, explain if each element of the sample space is equally likely to occur.a There are 150 000 tickets in a lottery. One ticket is drawn to win first prize.b There are twelve teams contesting a hockey tournament. One team is to win the

tournament.c A letter is chosen from the page of a book.

18 A poker machine has five wheels. Each wheel has 15 symbols on it. In how many ways can the wheels land?

19 There are four roads that lead from town A to town B, and five roads that lead from town B to town C. In how many different ways can I travel from town A to town C?

20 The Daily Double requires a punter to select the winner of two races. How many selections are possible if there are 16 horses in the first leg and 17 in the second leg?

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C h a p t e r 1 2 I n t r o d u c t i o n t o p r o b a b i l i t y 55321 At a restaurant, a patron has the choice of five entrees, eight main courses and four desserts.

In how many ways can they choose their meal?

22 Jake owns a bike chain that has a combination lock with four wheels. Each wheel has 10 digits.a How many different combinations are possible?b Jake has forgotten his combination. He can remember that the first digit is 5, and the last

digit is odd. How many different combinations could he try, to discover the correct combination to his chain?

23 The dial to a safe consists of 100 numbers. To open the safe, you must turn the dial to each of four numbers that form the safe’s combination.a How many different combinations to the safe are possible?b How many different combinations are possible if no number can be used twice?

24 From every 100 televisions on a production line, two are found to be defective. If you choose a television at random, find the relative frequency of defective televisions.

25 It is found that 150 of every thousand 17-year-old drivers will be involved in an accident within one year of having their driver’s licence.a What is the relative frequency of a 17-year-old driver having an accident?b If the average cost to an insurance company of each accident is $5000, what would be

the minimum premium that an insurance company should charge a 17-year-old driver?

26 The numbers 1 to 5 are written on the back of 5 cards that are turned face down. Michelle then chooses one card at random. Michelle wants to choose a number greater than 2. List the sample space and all favourable outcomes.

27 A barrel contains 25 numbered balls. One ball is drawn from the barrel. Find the probability that the marble drawn is:

28 A card is to be chosen from a standard deck. Find the probability that the card chosen is:

29 A video collection has 12 dramas, 14 comedies, 4 horror and 10 romance movies. If I choose a movie at random from the collection, find the probability that the movie chosen is:

30 The digits 5, 7, 8 and 9 are written on cards. They are then arranged to form a four-digit number. Find the probability that the number formed is:

31 A raffle has 2000 tickets sold and has two prizes. Michelle buys five tickets. a Find the probability that Michelle wins 1st prize.b If Michelle wins 1st prize, what is the probability that she also wins 2nd prize?

a 13 b 7 c an odd numberd a square number e a prime number f a double-digit number.

a the 2 of clubs b any 2 c any clubd a black card e a court card f a non-picture card.

a a comedy b a horror c not romance.

a 7895 b odd c divisible by 5d greater than 7000 e less than 8000.

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32 A barrel contains marbles with the numbers 1 to 40 on them. If one marble is chosen at random find, as a decimal, the probability that the number drawn is:

33 A carton of soft drinks contains 12 cola, 8 orange and 4 lemonade drinks. If a can is chosen at random from the carton, find the probability, as a percentage, that the can chosen is:

34 If an event has a probability of , would the event be unlikely, fifty-fifty or probable?

35 When 400 cars are checked for a defect, it is found that 350 have the defect. If one is chosen at random from the batch, find the probability that it has the defect and hence describe the chance of the car having the defect.

36 State the event which is complementary to each of the following.a Tossing a coin that lands Tailsb Rolling a die and getting a number less than 5c Choosing a blue ball from a bag containing 4 blue balls, 5 red balls and 7 yellow balls

37 A barrel contains 20 marbles of which 6 are black. One marble is selected at random. Find the probability that the marble selected is:

38 The probability that a person must stop at a set of traffic lights is . What is the probability of not needing to stop at the lights?

39 On a bookshelf there are 25 books. Of these, seven are fiction. If one book is chosen at random, what is the probability that the book chosen is non-fiction?

a 26 b even c greater than 10.

a cola b orange c not orange.

a black b not black.

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